JournalscmhVol. 82 , No. 1DOI 10.4171/cmh/87

The barycenter method on singular spaces

  • Peter Storm

    University of Pennsylvania, Philadelphia, United States
The barycenter method on singular spaces cover

Abstract

Compact convex cores with totally geodesic boundary are proven to uniquely minimize volume over all hyperbolic 3-manifolds in the same homotopy class. This solves a conjecture in Kleinian groups concerning acylindrical 3-manifolds. Closed hyperbolic manifolds are proven to uniquely minimize volume over all compact hyperbolic cone-manifolds in the same homotopy class with cone angles ≤2π. Closed hyperbolic manifolds are proven to minimize volume over all compact Alexandrov spaces with curvature bounded below by −1 in the same homotopy class. A version of the Besson–Courtois–Gallot theorem is proven for n-manifolds with boundary. The proofs extend the techniques of Besson–Courtois–Gallot.